Problem 62 - Precalculus Essentials

Question

Describe one similarity and one difference between the definitions of \(\sin \theta\) and \(\cos \theta,\) where \(\theta\) is an acute angle of a right triangle.

Step-by-Step Solution

Answer

The similarity between \( \sin \theta \) and \( \cos \theta \) is that both are ratios of sides of a right triangle and both lie between -1 and 1, inclusive. The difference is that, for \( \sin \theta \), the ratio is between the side opposite the angle and the hypotenuse, whereas for \( \cos \theta \), the ratio is between the side adjacent to the angle and the hypotenuse.

Step 1: Define Sin and Cos

In a right triangle, let \( \theta \) be an acute angle. The sine of \( \theta \) (written as \( \sin \theta \)) is defined as the ratio of the length of the side opposite \( \theta \) to the length of the hypotenuse. Similarly, the cosine of \( \theta \) (written as \( \cos \theta \)) is defined as the ratio of the length of the side adjacent to \( \theta \) to the length of the hypotenuse.

Step 2: Find Similarities

Both sine and cosine of an angle in a right triangle are ratios of sides of the triangle. They both are dependent on the angle \( \theta \) and both are values between -1 and 1 inclusively.

Step 3: Find Differences

The difference between the definitions of sin and cos lies in the sides of the triangle used in the ratios. While the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse, the cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.